Functional Graphs

There are some graphs of basic functions that every math student should know. If you can recognize what a function is going to look like when it is graphed on the xy plane, you’ll be a lot more likely to be able to figure out problems about that function, because you’ll be familiar with is.

Think of this – when someone is talking to you about a person, it’s hard to understand exactly what they are taking about, unless you have some kind of mental picture of the person they are talking about. Otherwise, you’ll be saying “Huh?”

So, to help you avoid saying “huh”, I am going to help you learn to recognize the graphs of basic functions.

First, , the most basic function. And probably the most boring. Blah.

And the next one, , this is a “parabola” – think “bowl.”

 

After the parabola, we must have a “cubic” (think, a “cube” has three dimensions – height, width, length)

Next, , also known as the “absolute value function”.

The slightly odd function, - a tricky function, because it is not defined at x=0!

Now for one of my very favorite functions,(an “exponential” with the famous number “e” – yes, it is a number.)

 

 

And we have have the exponential function without it’s inverse,(which is also not defined at x=0)

So there you have it – the seven basic functions (that are not trig functions – we’ll deal with those in another post). If you can memorize what these look like, you’ll be set through almost all of your math years. But I won’t let you get away with just memorizing them. Eventually we’ll study exactly why these functions look the way they do, and we’ll learn a lot of really amazing stuff about them.

Which is your favorite graph? Why?

(ps – I made these graphs using an amazing FREE graph drawing tool for your computer called WinPlot – if you don’t have a graphing calculator, this will be very useful for you. You can do a lot of the same things you can do on a graphing calculator using this program)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Core Standards – what your child should know

What should a child know?

You can always use your state’s curriculum guidelines for a starting point of what your child should know at what grade level, but if you are homeschooling, or you just want a general sense of what math topics are out there, you can try some national cores that have been developed by different organizations.

First, there are the Standards and Focal Points developed by the National Council of Teachers of Math.

There are also Mathematics standards published by the Common Core State Standards Initiative.

Both look pretty useful and in depth, and I hope to use them myself in coming up with topics to write about on this blog, as well as in teaching my own children.

What standards do you use in your math teaching? Do you look at your state core? If you homeschool, which standards do you use?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

you know more math than you think

Do you ever feel like math is a foreign language?

Think about how it feels to learn another language, or to be in a place where people are speaking a language you don’t understand.

Now think about how a child learns to speak their mother tongue (the language their parents speak). If a child is exposed to a language from the time they are born, and if someone interacts with them on a regular basis in that language, the child will eventually learn to speak that language. Are children born speaking a language? Maybe baby talk – but mostly they just cry.

Turns out, the ability to learn math is programmed in us similar to the way the ability to learn a language is programmed in us.

You can read about a study that was done on this subject here.

In the article, they mentioned that a lot of Euclidean Geometry (lines, space, shapes – all that basic stuff) concepts were “innate” in members of a remote tribal community in the Amazon who had no formal training in geometry. They got as many answers right about geometry as formally educated Americans.

Something that was interesting, however, was that younger children (5 and 6 years old) didn’t know the answers. The researchers weren’t sure why, but I have a hypothesis.

Since math is similar to a foreign language, and young children don’t often understand the concepts of grammar, it make sense that they also wouldn’t understand the concepts of math especially if math has not been “spoken” to them regularly. This goes back to the principle of teaching “pre-math” skills by exposing young children to concepts like classifying, sorting, number conservation, counting, etc.

If you want your child to understand math concepts better, speak math with them! Try to make it a point to “speak math” with your child every day. Whether it is by sorting and classifying, or by talking about shapes around you, or multiplying a recipe, or talking about how “steep” the stairs are, speak math with the children in your life!

How do you “speak math” with the children in your life?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Pre-Math: One of These Things

Sometimes as parents, and as teachers, we discount the things that children do as they play that help them learn math skills. We can encourage children to do things that develop good “math brains” without making them sit down and look at flash cards or do worksheets (I know a lot of kids love flash cards and worksheets, but we need to expand our horizons a little!)

One very important math concept and skill is that of sorting. In a previous post about combining like terms, I talked about how combining like terms was a lot like sorting laundry or marbles, or anything else than can be sorted and grouped.

Children like to sort and group naturally. My son will sort his toy cars by color, my daughter loves to help put the dishes in their correct places.

When our children practice sorting during play, or while helping mom and dad around the house, they are learning valuable math skills that will play a part in whether or not they are easily able to grasp the concept of things like combining like terms later in school.

Another part of sorting is being able to classify objects. In the picture above where my children are sorting laundry, they wouldn’t be able to sort the laundry unless they knew what set of classifications we were using to sort. Are we sorting kid’s clothes and grown-up’s clothes? Are we sorting every color? In our case, we were sorting according to “light” colors, “dark” and “ bright” colors, and “whites” – which meant that I had to explain what each of those classifiers meant.

When my children understood the classifiers, they were able to sort. Being able to classify objects (especially in different ways – like I said above) is also very significant and an important ability for children to develop so they will be successful in math skills later on. They have to be able to classify math “objects” such as numbers, coefficients, variables, exponents, etc.

Another game to play is “One of these things is not like the others” where you have a group of objects, and one object is not like the others. In this game, you are classifying objects, but you are also teaching the concept of “one” (which might seem like an easy concept, but trust me – you probably don’t actually understand “one”, even as an adult. I didn’t fully understand what “one” meant until I was finished with my undergraduate studies).

Try to find ways to help your children classify and sort objects. Try to make it a fun game that has to do with your life (sorting silverware or laundry, classifying cars when you are driving “Can you see a red car? How about a red truck? What about a blue truck?”). As your children learn to classify and sort objects, and as they do it on a daily basis, their brains will be gearing up to understand much more complex mathematical situations as they grow up.

What ways do you teach your children how to classify and sort? Do you feel like you are able to teach your children math concepts?

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Comparing Decimals

Today, Student A had a worksheet for comparing decimals. She was supposed to say which decimal in each pair was greater than the other, or if they were equal.

Because she originally completed the assignment by guessed (a pretty usual occurrence for A), I decided to give her some steps to follow in comparing decimals before we even began correcting the assignment.

First, we talked about how decimals are just like fractions – meaning they are just part of a number, instead of a whole number. With that in mind, we went on to our steps for comparing decimals:

Step 1. If there is a whole number (the part of the number that is to the left of the decimal point), compare the whole numbers first. If the whole numbers are different, then all you have to do is tell which whole number is greater. (for example, 12.3 and 14.72 – 14.72 is greater because 14 > 12, and it doesn’t matter how much “extra” you tack on to the 12, 12 is still smaller than 14). Otherwise, move on to step 2.
Step 2. Since the whole number parts are the same, we will need to compare the decimals. Remember that decimals are like fractions, so .1 = 1/10, and .01 = 1/100. Well, it’s hard to compare fractions when the bottoms are different (that is, it’s not very intuitive) and it is equally hard to compare decimals when we don’t have the same number of places, such as in 4.35 and 4.3.
To make things easier to “see” (instead of having to think too hard – mathematicians never like to think harder than they absolutely have to – that’s why we have “tricks” for everything) we are going to make sure each decimal has the same number of “places”. To do this, all we have to do is add zeros on the right hand side of the number. Going back to our example of .35 and .3, since .35 has two places and .3 only has one, we will add a zero to .3 and have .30. Now we are comparing .30 and .35 – well, that is just like comparing 30 and 35. It is clear that 35 > 30, so .35 > .30.
If you had .30 and .3, when you add a zero to .3 you will get .30 and .30 – in this case, the numbers are equal.

I was really impressed with Student A’s internalizing this concept. After five or six problems she was able to figure them out quickly, and explain her answers just as fast. She really understood the concept, and that was a huge success for us.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Times on the Table

One of the things I worked with a lot with student A over the summer was TIMES TABLES.

While I am not a supporter of a lot of rote repetition when it comes to learning math, I definitely support repetition when it comes to remembering math. But only after the initial learning has taken place.

It doesn’t do a student any good to memorize the times tables if the student doesn’t even understand what the times tables are. Such was the case with A. We spent a lot of time just learning about numbers in general, and then specifically multiplication, before we moved on to memorizing the times tables.

As part of “figuring out” the times tables, we learned a lot of tricks, and used some games, songs, and pictures to learn the times tables. We looked at patterns in the numbers, talked about sets, and played a lot with Cuisenaire rods to get a physical grasp of the concept.

Only then did we move on to drills.

Here are the posts describing how we learned each set of times tables.

Twos
Threes – Three is a Magic Number
Fours
Fives
Sixes
Sevens
Eights
Nines – The Great Nines
Tens
Elevens – Some Tricky 11’s – in each place
Twelves

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Combining Like Terms – Sorting Letters

Did you ever play sorting games as a little kid? If you fold your own laundry, then you should be a pro at sorting – pants, shirts, socks, sweaters.

Combing like terms is just like sorting laundry, or marbles (as illustrated in my sorry attempt at drawing marbles).

When we have “letters” (also known as variables) in an expression we often want to combine them so that we have an easier expression to look at.

Think of it like having a pile full of different color marbles (like in the illustration above). Let’s look at an example:

2x + 4y – 3xy + 7y – 5x + 2xy

Let’s look at the different “color” marbles we have (that is, the different “terms”)

We have x’s, y’s, and xy’s (the xy’s are their own type of term, because xy is not the same as x or y).

Now we combine the things that are “like” – We have 2x and –5x, so that is –3x. We have 4y and 7y, so that is 11y. Finally, we have –3xy and 2xy, which is –xy.

Our “simplified” expression is –3x +11y –xy.

Because of the commutative property of addition, we can write the expression in any order we want to.

-3x – xy + 11y
11y – 3x – xy

etc, etc

Here are some practice problems for combining like terms (also known as “simplifying expressions”)

Simplify the Expressions by combining like terms

1.

2.

3.

4.

5.

(Pre-Math Tip! Play sorting games with your preschooler. You may not realize it now when you are sorting blocks and dishes and beads, but your preschooler is building the foundation for basic algebra skills like being able to combine like terms!)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.